88504
The largest interval for which \(x^{12}-x^{9}+x^{4}-x\) \(+\mathbf{1}>\mathbf{0}\) is
1 \(-4\lt x\lt 0\)
2 \(0\lt x\lt 1\)
3 \(-100\lt \mathrm{x}\lt 100\)
4 \(-\infty\lt x\lt \infty\)
Explanation:
(D) : Given, \(\mathrm{x}^{12}-\mathrm{x}^{9}+\mathrm{x}^{4}-\mathrm{x}+1>0\) Let us consider \(f(x)=x^{12}=x^{12}-x^{9}+x^{4}-x+1\) \(=\mathrm{x}^{9}\left(\mathrm{x}^{3}-1\right)+\mathrm{x}\left(\mathrm{x}^{3}-1\right)+1\) \(=\left(\mathrm{x}^{9}+\mathrm{x}\right)\left(\mathrm{x}^{3}-1\right)+1>0 \forall \mathrm{x} \geq 1\) After this \(f(x)=x^{12}-x^{9}+x^{4}-x+1=x^{4}\left(x^{8}+1\right)-x\left(x^{8}+1\right)+1\) \(=\left(\mathrm{x}^{8}+1\right)\left(\mathrm{x}^{4}-\mathrm{x}\right)+1\) \(=x\left(x^{8}+1\right)\left(x^{3}-1\right)+1>0 \forall x \leq 0 \tag{ii}\) Again, \(f(x)=x^{12}-x^{9}+x^{4}-x+1 \tag{iii}\) \(=x^{12}+x^{4}\left(1-x^{5}\right)+(1-x)>0\) for \(0\lt x\lt 1\) Combining equation (i), (ii) and (iii), we get - \(f(x)>0 \text { for } x \in(-\infty, \infty) \text {. }\)
Jamia Millia Islamia-2009
Linear Inequalities and Linear Programming
88505
The values of \(x\) for which \(4^{x}+4^{1-x}-5\lt 0\), is given by
88510
For a real number \(r\) we denote by \([r]\) the largest integer less than or equal to \(r\). If \(x, y\) are real numbers with \(x, y \geq 1\) then which of the following statements is always true?
88504
The largest interval for which \(x^{12}-x^{9}+x^{4}-x\) \(+\mathbf{1}>\mathbf{0}\) is
1 \(-4\lt x\lt 0\)
2 \(0\lt x\lt 1\)
3 \(-100\lt \mathrm{x}\lt 100\)
4 \(-\infty\lt x\lt \infty\)
Explanation:
(D) : Given, \(\mathrm{x}^{12}-\mathrm{x}^{9}+\mathrm{x}^{4}-\mathrm{x}+1>0\) Let us consider \(f(x)=x^{12}=x^{12}-x^{9}+x^{4}-x+1\) \(=\mathrm{x}^{9}\left(\mathrm{x}^{3}-1\right)+\mathrm{x}\left(\mathrm{x}^{3}-1\right)+1\) \(=\left(\mathrm{x}^{9}+\mathrm{x}\right)\left(\mathrm{x}^{3}-1\right)+1>0 \forall \mathrm{x} \geq 1\) After this \(f(x)=x^{12}-x^{9}+x^{4}-x+1=x^{4}\left(x^{8}+1\right)-x\left(x^{8}+1\right)+1\) \(=\left(\mathrm{x}^{8}+1\right)\left(\mathrm{x}^{4}-\mathrm{x}\right)+1\) \(=x\left(x^{8}+1\right)\left(x^{3}-1\right)+1>0 \forall x \leq 0 \tag{ii}\) Again, \(f(x)=x^{12}-x^{9}+x^{4}-x+1 \tag{iii}\) \(=x^{12}+x^{4}\left(1-x^{5}\right)+(1-x)>0\) for \(0\lt x\lt 1\) Combining equation (i), (ii) and (iii), we get - \(f(x)>0 \text { for } x \in(-\infty, \infty) \text {. }\)
Jamia Millia Islamia-2009
Linear Inequalities and Linear Programming
88505
The values of \(x\) for which \(4^{x}+4^{1-x}-5\lt 0\), is given by
88510
For a real number \(r\) we denote by \([r]\) the largest integer less than or equal to \(r\). If \(x, y\) are real numbers with \(x, y \geq 1\) then which of the following statements is always true?
88504
The largest interval for which \(x^{12}-x^{9}+x^{4}-x\) \(+\mathbf{1}>\mathbf{0}\) is
1 \(-4\lt x\lt 0\)
2 \(0\lt x\lt 1\)
3 \(-100\lt \mathrm{x}\lt 100\)
4 \(-\infty\lt x\lt \infty\)
Explanation:
(D) : Given, \(\mathrm{x}^{12}-\mathrm{x}^{9}+\mathrm{x}^{4}-\mathrm{x}+1>0\) Let us consider \(f(x)=x^{12}=x^{12}-x^{9}+x^{4}-x+1\) \(=\mathrm{x}^{9}\left(\mathrm{x}^{3}-1\right)+\mathrm{x}\left(\mathrm{x}^{3}-1\right)+1\) \(=\left(\mathrm{x}^{9}+\mathrm{x}\right)\left(\mathrm{x}^{3}-1\right)+1>0 \forall \mathrm{x} \geq 1\) After this \(f(x)=x^{12}-x^{9}+x^{4}-x+1=x^{4}\left(x^{8}+1\right)-x\left(x^{8}+1\right)+1\) \(=\left(\mathrm{x}^{8}+1\right)\left(\mathrm{x}^{4}-\mathrm{x}\right)+1\) \(=x\left(x^{8}+1\right)\left(x^{3}-1\right)+1>0 \forall x \leq 0 \tag{ii}\) Again, \(f(x)=x^{12}-x^{9}+x^{4}-x+1 \tag{iii}\) \(=x^{12}+x^{4}\left(1-x^{5}\right)+(1-x)>0\) for \(0\lt x\lt 1\) Combining equation (i), (ii) and (iii), we get - \(f(x)>0 \text { for } x \in(-\infty, \infty) \text {. }\)
Jamia Millia Islamia-2009
Linear Inequalities and Linear Programming
88505
The values of \(x\) for which \(4^{x}+4^{1-x}-5\lt 0\), is given by
88510
For a real number \(r\) we denote by \([r]\) the largest integer less than or equal to \(r\). If \(x, y\) are real numbers with \(x, y \geq 1\) then which of the following statements is always true?
88504
The largest interval for which \(x^{12}-x^{9}+x^{4}-x\) \(+\mathbf{1}>\mathbf{0}\) is
1 \(-4\lt x\lt 0\)
2 \(0\lt x\lt 1\)
3 \(-100\lt \mathrm{x}\lt 100\)
4 \(-\infty\lt x\lt \infty\)
Explanation:
(D) : Given, \(\mathrm{x}^{12}-\mathrm{x}^{9}+\mathrm{x}^{4}-\mathrm{x}+1>0\) Let us consider \(f(x)=x^{12}=x^{12}-x^{9}+x^{4}-x+1\) \(=\mathrm{x}^{9}\left(\mathrm{x}^{3}-1\right)+\mathrm{x}\left(\mathrm{x}^{3}-1\right)+1\) \(=\left(\mathrm{x}^{9}+\mathrm{x}\right)\left(\mathrm{x}^{3}-1\right)+1>0 \forall \mathrm{x} \geq 1\) After this \(f(x)=x^{12}-x^{9}+x^{4}-x+1=x^{4}\left(x^{8}+1\right)-x\left(x^{8}+1\right)+1\) \(=\left(\mathrm{x}^{8}+1\right)\left(\mathrm{x}^{4}-\mathrm{x}\right)+1\) \(=x\left(x^{8}+1\right)\left(x^{3}-1\right)+1>0 \forall x \leq 0 \tag{ii}\) Again, \(f(x)=x^{12}-x^{9}+x^{4}-x+1 \tag{iii}\) \(=x^{12}+x^{4}\left(1-x^{5}\right)+(1-x)>0\) for \(0\lt x\lt 1\) Combining equation (i), (ii) and (iii), we get - \(f(x)>0 \text { for } x \in(-\infty, \infty) \text {. }\)
Jamia Millia Islamia-2009
Linear Inequalities and Linear Programming
88505
The values of \(x\) for which \(4^{x}+4^{1-x}-5\lt 0\), is given by
88510
For a real number \(r\) we denote by \([r]\) the largest integer less than or equal to \(r\). If \(x, y\) are real numbers with \(x, y \geq 1\) then which of the following statements is always true?
